Problem 35 - Algebra and Trigonometry

Question

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ \frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \frac{1}{3^{4}}, \dots $$

Step-by-Step Solution

Answer

The first six partial sums are \( S_1 = \frac{1}{3} \), \( S_2 = \frac{4}{9} \), \( S_3 = \frac{13}{27} \), \( S_4 = \frac{40}{81} \), \( S_5 = \frac{121}{243} \), \( S_6 = \frac{364}{729} \).

Step 1: Understanding the Sequence

The given sequence is \( \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, \ldots \). It is a geometric sequence with the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \).

Step 2: Define Partial Sum

The \( n^{th} \) partial sum \( S_n \) is the sum of the first \( n \) terms of the sequence. The formula for the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = a \frac{1-r^n}{1-r} \] where \( a \) is the first term and \( r \) is the common ratio.

Step 3: Calculate \( S_1 \)

For \( S_1 \), there is only the first term: \[ S_1 = \frac{1}{3} \]

Step 4: Calculate \( S_2 \)

For \( S_2 \), sum of the first two terms: \[ S_2 = \frac{1}{3} + \frac{1}{3^2} = \frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9} \]

Step 5: Calculate \( S_3 \)

For \( S_3 \), sum of the first three terms: \[ S_3 = \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} = \frac{4}{9} + \frac{1}{27} = \frac{12}{27} + \frac{1}{27} = \frac{13}{27} \]

Step 6: Calculate \( S_4 \)

For \( S_4 \), sum of the first four terms: \[ S_4 = \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} = \frac{13}{27} + \frac{1}{81} = \frac{39}{81} + \frac{1}{81} = \frac{40}{81} \]

Step 7: Calculate \( S_5 \)

For \( S_5 \), sum of the first five terms: \[ S_5 = \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} + \frac{1}{3^5} = \frac{40}{81} + \frac{1}{243} = \frac{120}{243} + \frac{1}{243} = \frac{121}{243} \]

Step 8: Calculate \( S_6 \)

For \( S_6 \), sum of the first six terms: \[ S_6 = \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} + \frac{1}{3^5} + \frac{1}{3^6} = \frac{121}{243} + \frac{1}{729} = \frac{363}{729} + \frac{1}{729} = \frac{364}{729} \]

Key Concepts

Geometric Sequence

A geometric sequence is a specific type of sequence where each term is found by multiplying the previous term by a constant number, called the common ratio. This sequence grows or shrinks constantly by this fixed ratio. Consider the sequence: \( \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, \ldots \). Here, each term is arrived at by multiplying the previous term by the common ratio \( \frac{1}{3} \). This characteristic of having a constant multiplier makes it a geometric sequence. Understanding this concept is fundamental when analyzing the behavior and properties of the sequence. It determines how quickly the sequence converges or diverges, depending on whether the common ratio is less than or greater than 1.

Common Ratio

The common ratio in a geometric sequence is the consistent factor that you multiply with to obtain the next term. In the given sequence \( \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \ldots \), the common ratio \( r \) is \( \frac{1}{3} \). This means every successive term in the sequence is \( \frac{1}{3} \) times the previous term.
To identify the common ratio, it’s key formula is: - Choose any term in the sequence (except the first one) and divide it by the preceding term.For our current sequence, dividing the second term \( \frac{1}{9} \) by the first term \( \frac{1}{3} \) provides us the common ratio: \( \frac{1}{3} \). Simply knowing the common ratio allows us to calculate future terms and sum of terms efficiently.

First Term

In a geometric sequence, the first term is crucial as it forms the basis from which the entire sequence is generated. In our example, the first term \( a \) is \( \frac{1}{3} \). It is often denoted by \( a_1 \) in mathematics.
Here's why the first term is important:- It serves as a starting point for the sequence.- It is used in the formula for calculating the \( n^{th} \) partial sum, which is given by \[ S_n = a \frac{1-r^n}{1-r} \].The first term influences the values of all subsequent terms in the sequence, as each term is a product of the first term and the common ratio raised to some power. Thus, understanding the first term helps in predicting the nature and behavior of the sequence. Knowing this allows us to calculate sums and understand the progression better.